M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 15
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The 13 semi-regular polyhedra are listed in Table 2-4 and someof them are depicted in Figure 2-59. Table 2-4 also enumerates theirrotation axes.The simplest semi-regular polyhedra are obtained by symmetrically shaving off the corners of the regular solids. They are the truncated regular polyhedra and are marked with the superscript “a” inTable 2-4.
One of them is the truncated icosahedron, the shape of2.8. Polyhedra89Figure 2-60. Shapes and various modes of linkage of truncated octahedral-shapedsodalite units [97]. Used by permission from Plenum Press and Brian Beagley.the C60 buckminsterfullerene molecule. Two of the semi-regular polyhedra are classified as so-called quasi-regular polyhedra. They havetwo kinds of faces, and each face of one kind is entirely surroundedby faces of the other kind. They are marked with the superscript “b”in Table 2-4.
All these special 7 semi-regular polyhedra are shownin Figure 2-59. The remaining six semi-regular polyhedra may bederived from the other semi-regular polyhedra.The structures of zeolites, aluminosilicates, are rich in polyhedralshapes, including the channels and cavities they form [95]. One ofthe most common zeolites is sodalite, Na6 [Al6 Si6 O24 ]·2 NaCl, whosename refers to its sodium content. The sodalite unit has the shape ofa truncated octahedron.
The three models of Figure 2-60 representdifferent modes of linkages between the sodalite units. It is especiallyinteresting to see the different cavities formed by different modes oflinkage [96]. Two artistic representations of semi-regular polyhedraare shown in Figure 2-61.The prisms and antiprisms are also important polyhedron families.A prism has two congruent and parallel faces and they are joinedby a set of parallelograms. An antiprism also has two congruentand parallel faces but they are joined by a set of triangles. An infinite number of prisms and antiprisms exist and a few are shown inFigure 2-62.
A prism or an antiprism is semiregular if all its faces are902 Simple and Combined Symmetries(a)(b)Figure 2-61. Two artistic representations of semi-regular polyhedra (photographsby the authors). (a) Snub cube fountain in Pasadena, California [98]; (b) Cuboctahedron on top of a garden lantern in the Shugakuin Imperial Villa in Kyoto [99].regular polygons. A cube can be considered a square prism, and anoctahedron a triangular antiprism.There are many additional polyhedra that are important indiscussing molecular geometries and crystal structures.
SantiagoAlvarez compiled an extensive discussion of polyhedra of varyingregularity, their relationships, and many examples from inorganicchemistry [100]. We will return to this topic in the next Chapter.Figure 2-62. Prisms and antiprisms.References91References1. G. H. Hardy, A Mathematician’s Apology, Cambridge University Press,Cambridge, 1941.2. T. Mann, The Magic Mountain. The cited passage is in French both in the original German (see, e.g., T. Mann, Der Zauberberg.
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